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MathematicsList match+4 / -1Question 15 of 48

JEE Advanced 2026 · Paper 1, Mathematics question 15

For real numbers α, β, γ, δ\alpha,\ \beta,\ \gamma,\ \delta and μ\mu, consider the matrix
M=[α12−1213β13γδμ].M=\begin{bmatrix}\alpha&\frac1{\sqrt2}&-\frac1{\sqrt2}\\\frac1{\sqrt3}&\beta&\frac1{\sqrt3}\\\gamma&\delta&\mu\end{bmatrix}.
Suppose that MMT=IMM^T=I, where MTM^T is the transpose of the matrix MM, and II is the 3×33\times3 identity matrix. Let
u⃗=α i^+13j^+γ k^,v⃗=12i^+βj^+δk^andw⃗=−12i^+13j^+μk^.\vec u=\alpha\,\hat i+\frac1{\sqrt3}\hat j+\gamma\,\hat k,\qquad\vec v=\frac1{\sqrt2}\hat i+\beta\hat j+\delta\hat k\quad\text{and}\quad\vec w=-\frac1{\sqrt2}\hat i+\frac1{\sqrt3}\hat j+\mu\hat k.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I

  • P
    The value of γ2+δ2\gamma^2+\delta^2 is
  • Q
    If xu⃗+yv⃗+zw⃗=j^x\vec u+y\vec v+z\vec w=\hat j for some real numbers x, yx,\ y and zz, then the value of xx is
  • R
    The value of ∣u⃗⋅(v⃗×w⃗)∣\left|\vec u\cdot\left(\vec v\times\vec w\right)\right| is
  • S
    The value of ∣u⃗×(v⃗×w⃗)∣\left|\vec u\times\left(\vec v\times\vec w\right)\right| is

List-II

  • 1
    00
  • 2
    11
  • 3
    12\dfrac1{\sqrt2}
  • 4
    13\dfrac1{\sqrt3}
  • 5
    56\dfrac56
Choose one option
Answer and worked solution

Official answer: A

MMT=IMM^T=I makes MM orthogonal, so its rows and its columns are orthonormal. Note u⃗,v⃗,w⃗\vec u,\vec v,\vec w are its columns.

Entries.

  • Row 1: α2+12+12=1\alpha^2+\frac12+\frac12=1, so α=0\alpha=0.
  • Row 2: β2=13\beta^2=\frac13; row 1 · row 2 =β2−16=0=\frac{\beta}{\sqrt2}-\frac1{\sqrt6}=0, so β=13\beta=\frac1{\sqrt3}.
  • Column 1: 13+γ2=1\frac13+\gamma^2=1, so γ2=23\gamma^2=\frac23. Column 2: 12+13+δ2=1\frac12+\frac13+\delta^2=1, so δ2=16\delta^2=\frac16.

(P) γ2+δ2=56\gamma^2+\delta^2=\frac56 → 5.

(Q) The columns are orthonormal, so x=u⃗⋅j^=13x=\vec u\cdot\hat j=\frac1{\sqrt3} → 4.

(R) ∣u⃗⋅(v⃗×w⃗)∣=∣det⁡M∣=1|\vec u\cdot(\vec v\times\vec w)|=|\det M|=1 → 2.

(S) v⃗×w⃗=±u⃗\vec v\times\vec w=\pm\vec u, so u⃗×(v⃗×w⃗)=0⃗\vec u\times(\vec v\times\vec w)=\vec0 → 1.

P→5, Q→4, R→2, S→1.

Topics:Special matricesScalar and vector triple products
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